Recognizing, of course the Latin language as the basis for the Romantic languages. As Rome expanded, did the term "Latin" follow them? If so, why was it jettisoned as a descriptor for Turkish, Italian, French, German, North African, Eastern European, Persian and Semitic peoples and not Iberian peoples?
edit: grammar.
I think your question might be based on false premises. Many people whose mother tongue is a Romance language have been and/or are called "Latin." For example, since the 19th century it is not uncommon to see Romanians described as Latins, many of whom self-identify as Latin.
I think what you're referring to is prevalence of the term "Latin America," used to describe the countries south of the USA, which just so happen to (mostly) be Spanish- and Portguese-speaking countries. The term Latin America first became popularized in the mid-19th century, when Napoleon III began to use it to imply an ancient bond between South and Central America and the French Empire. Of course, in turn he was intending to further his imperial projects in the region, such as installing Maximilian as the Emperor of Mexico. While the term Latin America has come to usually mean the Spanish- and Portuguese-speaking countries of the Americas, it is not restricted to them; and still many people include such countries as Haiti, French Guyana, and sometimes even the province of Quebec in Latin America.
Source: Chasteen, John Charles. Born in Blood and Fire: A Concise History of Latin America.
Edit: grammar.
Would religion also contribute to this? I've hear Catholics referred to as 'Latins'